Properties

Label 2-4002-1.1-c1-0-33
Degree $2$
Conductor $4002$
Sign $1$
Analytic cond. $31.9561$
Root an. cond. $5.65297$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 2.56·5-s − 6-s + 3.26·7-s − 8-s + 9-s − 2.56·10-s + 0.513·11-s + 12-s − 3.20·13-s − 3.26·14-s + 2.56·15-s + 16-s − 2.52·17-s − 18-s + 3.58·19-s + 2.56·20-s + 3.26·21-s − 0.513·22-s + 23-s − 24-s + 1.56·25-s + 3.20·26-s + 27-s + 3.26·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.14·5-s − 0.408·6-s + 1.23·7-s − 0.353·8-s + 0.333·9-s − 0.810·10-s + 0.154·11-s + 0.288·12-s − 0.889·13-s − 0.871·14-s + 0.661·15-s + 0.250·16-s − 0.611·17-s − 0.235·18-s + 0.821·19-s + 0.572·20-s + 0.711·21-s − 0.109·22-s + 0.208·23-s − 0.204·24-s + 0.312·25-s + 0.628·26-s + 0.192·27-s + 0.616·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4002\)    =    \(2 \cdot 3 \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(31.9561\)
Root analytic conductor: \(5.65297\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4002,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.541595469\)
\(L(\frac12)\) \(\approx\) \(2.541595469\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
good5 \( 1 - 2.56T + 5T^{2} \)
7 \( 1 - 3.26T + 7T^{2} \)
11 \( 1 - 0.513T + 11T^{2} \)
13 \( 1 + 3.20T + 13T^{2} \)
17 \( 1 + 2.52T + 17T^{2} \)
19 \( 1 - 3.58T + 19T^{2} \)
31 \( 1 + 9.93T + 31T^{2} \)
37 \( 1 - 4.60T + 37T^{2} \)
41 \( 1 - 6.66T + 41T^{2} \)
43 \( 1 - 1.43T + 43T^{2} \)
47 \( 1 - 6.20T + 47T^{2} \)
53 \( 1 - 6.04T + 53T^{2} \)
59 \( 1 - 8.70T + 59T^{2} \)
61 \( 1 - 0.354T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 + 8.47T + 83T^{2} \)
89 \( 1 - 2.85T + 89T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.616951766616625472875110681169, −7.61123147933625643433077490108, −7.37715387821468825447172941392, −6.32363335781910187642696462101, −5.46598733147005851199298878677, −4.83884406080043666699980820402, −3.74777876793980576978771105976, −2.42266334594031928169210101362, −2.07737583959474543666899290444, −1.04303838184441728380202124826, 1.04303838184441728380202124826, 2.07737583959474543666899290444, 2.42266334594031928169210101362, 3.74777876793980576978771105976, 4.83884406080043666699980820402, 5.46598733147005851199298878677, 6.32363335781910187642696462101, 7.37715387821468825447172941392, 7.61123147933625643433077490108, 8.616951766616625472875110681169

Graph of the $Z$-function along the critical line